Question

In ΔABC, AB = AC and the bisectors of angles B and C intersect at point O.Prove that BO = CO and the ray AO is the bisector of angle BAC.

Answer 1

In ΔABC,
Since AB = AC
∠C = ∠B    ...(angles opposite to the equal sides are equal)
BO and CO are angle bisectors of ∠B and ∠C respectively
Hence, ∠ABO = ∠OBC = ∠BCO = ∠ACO
Join AO to meet BC at D
In ΔABO and ΔACO and 
AO = AO
AB = AC
∠C = ∠B = 
Therefore, ΔBAO ≅ ΔACO   ...(SAS criteria)
Hence, ∠BAO = ∠CAO
⇒ AO bisects angle BAC
In ΔABO and ΔACO
and AB = AC
AO = AO
∠BAD = ∠CAD =  ...(proved)
ΔBAO ≅ ΔACO       ...(SAS criteria)
Therefore,
BO = CO.